Engraving a 2d image on a subdivision surface

ABSTRACT

It is the method comprising the steps of defining, by a user, a base mesh associated to a subdivision surface and to a corresponding predetermined mesh-to-NURBS-surface conversion algorithm, the subdivision surface representing the 3D modeled object; defining, by the user, a 2D image and a location for engraving the 2D image on the subdivision surface; and determining a NURBS surface that corresponds to applying a deformation map on the result of performing the mesh-to-NURBS-surface conversion algorithm to the base mesh, the deformation map including displacement vectors provided for positions of the result of performing the mesh-to-NURBS-surface conversion algorithm to the base mesh, the positions corresponding to the location for engraving the 2D image, the displacement vectors being computed based on corresponding pixel values of the 2D image. Such a method improves the design of a 3D modeled object.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119 or 365 to EuropeanApplication No. 15305153.7, filed Feb. 2, 2015. The entire teachings ofthe above application(s) are incorporated herein by reference.

FIELD OF THE INVENTION

The invention notably relates to the field of computer-aided design(CAD), and more particularly to a method, program and system fordesigning a 3D modeled object by graphical user-interaction, with a dataprocessing based on a 2D image.

BACKGROUND

A number of systems and programs are offered on the market for thedesign, the engineering and the manufacturing of objects. CAD is anacronym for Computer-Aided Design, e.g. it relates to software solutionsfor designing an object. CAE is an acronym for Computer-AidedEngineering, e.g. it relates to software solutions for simulating thephysical behavior of a future product. CAM is an acronym forComputer-Aided Manufacturing, e.g. it relates to software solutions fordefining manufacturing processes and operations. In such computer-aideddesign systems, the graphical user interface plays an important role asregards the efficiency of the technique. These techniques may beembedded within Product Lifecycle Management (PLM) systems. PLM refersto a business strategy that helps companies to share product data, applycommon processes, and leverage corporate knowledge for the developmentof products from conception to the end of their life, across the conceptof extended enterprise.

The PLM solutions provided by DASSAULT SYSTEMES (under the trademarksCATIA, ENOVIA and DELMIA) provide an Engineering Hub, which organizesproduct engineering knowledge, a Manufacturing Hub, which managesmanufacturing engineering knowledge, and an Enterprise Hub which enablesenterprise integrations and connections into both the Engineering andManufacturing Hubs. All together the system delivers an open objectmodel linking products, processes, resources to enable dynamic,knowledge-based product creation and decision support that drivesoptimized product definition, manufacturing preparation, production andservice.

CAD solutions make extensive use of two main widely known technologiesfor surface modeling: Non Uniform Rational B-Spline surfaces (NURBS inthe following) and Subdivision surfaces.

NURBS technology is based on a polynomial or rational parameterizationhandled by rectangular grids of 3D points, named the “control points”.Edition of a NURBS surface is performed by moving control points. Thetechnology is designed in such a way that the degree of polynomials doesnot depend on the number of control points. Furthermore, the influenceof control points is restricted to a neighborhood. Consequently, theuser is able to locally modify the shape of the surface, the regionoutside of the said neighborhood being untouched. A classical referencein this field is textbook The NURBS book, L. Piegl and T. Wayne,Springer, 2^(nd) edition, 2013.

On the other hand, a subdivision surface is defined by a 3-dimensional“base mesh” made of points and line segments. The base mesh isuser-defined, and may feature an arbitrary topology, as opposed to therectangular grid of NURBS' control points. The base mesh governs theoverall shape of the subdivision surface and can be edited by the user.A change in the base mesh affects the whole subdivision surface. Asopposed to NURBS surfaces, the subdivision surface is a limit object ofa subdivision process starting with the base mesh. FIG. 1 illustrates athree holes closed base mesh 11 and the first subdivision steps. Thelimit surface is a smooth three holes-torus like object 13. Thefundamentals are explained notably in the following reference:Recursively generated B-Spline surfaces on arbitrary topological meshes,E. Catmull, J. Clark, Computer Aided Design, Vol. 10, No 6, November1978. It is noted that it is possible to approximate a subdivisionsurface by an arrangement of NURBS surfaces, with a conversion of thebase mesh into a NURBS surface, as explained in documents U.S. Pat. No.7,595,799 B2 (corresponding to EP 1750229 B1) and U.S. Pat. No.7,400,323 B2.

In short, NURBS technology features a local modification capability anda grid-defined topology and Subdivision technology features an arbitrarytopology and a non-local edition capability. Clearly, creation of smalldetails on a surface while saving its overall shape is well supported bythe NURBS technology because of its ability to handle localmodification. Conversely, this capability is not supported bySubdivision surface technology. On the other hand, arbitrary topology issupported by subdivision surface technology as opposed to NURBS surface,the topology of which being restricted to a rectangular grid of controlpoints.

What is missing in the prior art is notably the capability of a localchange on a surface with arbitrary topology. Within this context, thereis still a need for an improved solution to design a 3D modeled object.

SUMMARY OF THE INVENTION

It is therefore provided a computer-implemented method for designing a3D modeled object. The method comprises the step of defining, by a user,a base mesh associated to a subdivision surface and to a correspondingpredetermined mesh-to-NURBS-surface conversion algorithm. Thesubdivision surface represents the 3D modeled object. The method alsocomprises the step of defining, by the user, a 2D image and a locationfor engraving the 2D image on the subdivision surface. The method alsocomprises the step of determining a NURBS surface that corresponds toapplying a deformation map on the result of performing themesh-to-NURBS-surface conversion algorithm to the base mesh. Thedeformation map includes displacement vectors provided for positions ofthe result of performing the mesh-to-NURBS-surface conversion algorithmto the base mesh. The positions correspond to the location for engravingthe 2D image. The displacement vectors are computed based oncorresponding pixel values of the 2D image.

The method may comprise one or more of the following:

-   -   the method comprises computing the displacement vectors,        including determining, from the base mesh, a first 3D mesh that        corresponds to the location for engraving the 2D image,        determining, from the first 3D mesh, a second 3D mesh by        displacing vertices of the first 3D mesh, in a direction normal        to the first 3D mesh, based on the corresponding pixel values of        the 2D image, and computing, for positions of the result of        performing the mesh-to-NURBS-surface conversion algorithm to the        base mesh, a displacement vector equal to the 3D position        difference between the corresponding control point of the result        of performing the mesh-to-NURBS-surface conversion algorithm to        the second 3D mesh and the corresponding control point of the        result of performing the mesh-to-NURBS-surface conversion        algorithm to the first 3D mesh;    -   determining the first 3D mesh includes extracting a sub-mesh of        the base mesh that corresponds to the location for engraving the        2D image, and then subdividing the sub-mesh a predetermined        number of times;    -   the predetermined number of times is defined by the user and/or        increasingly depends on a predetermined criterion associated to        a level of details of the 2D image;    -   the corresponding pixel values of the 2D image are pixel        grayscale values;    -   the amplitude of a displacement vector increasingly depends on        the corresponding grayscale value; and/or    -   the user further selects an engraving direction from a        predetermined binary list.

It is further provided a 3D modeled object designed by the method.

It is further provided a computer file including specifications of the3D modeled object.

It is further provided a computer program comprising instructions forperforming the method.

It is further provided a computer readable data storage medium havingrecorded thereon the computer program.

It is further provided a CAD system comprising a processor coupled to amemory and a graphical user interface, the memory having recordedthereon the computer program.

It is further provided a method for manufacturing an industrial product,comprising the steps of designing a three-dimensional object thatrepresents the industrial product according to the above design method,and then manufacturing the industrial product based on the designedthree-dimensional object.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way ofnon-limiting example, and in reference to the accompanying drawings,where:

FIG. 1 shows a subdivision surface;

FIG. 2 flowchart of an example of the method;

FIG. 3 shows an example of a graphical user interface of the system;

FIG. 4 shows an example of the system; and

FIGS. 5-26 illustrate the method.

DETAILED DESCRIPTION OF THE INVENTION

With reference to the flowchart of FIG. 2, it is proposed acomputer-implemented method for designing a 3D modeled object. Themethod comprises a step of defining S10, by a user, a base meshassociated to a subdivision surface and to a corresponding predeterminedmesh-to-NURBS-surface conversion algorithm. The subdivision surfacerepresents the 3D modeled object. The method also comprises definingS20, by the user, a 2D image and a location for engraving the 2D imageon the subdivision surface. And the method then comprises determiningS30 a NURBS surface that corresponds (i.e. by “corresponds”, it is meantthat the specific way to obtain the NURBS surface is an implementationdetail, as long as the result would be obtained—at least approximately,e.g. in the meaning of “approximation” provided later—by the followingsteps, even if those specific steps are not implemented) to applying adeformation map on the result of performing the mesh-to-NURBS-surfaceconversion algorithm to the base mesh. The deformation map includesdisplacement vectors provided for positions of the result of performingthe mesh-to-NURBS-surface conversion algorithm to the base mesh. Thesepositions correspond to the location for engraving the 2D image. Thedisplacement vectors are computed based on corresponding pixel values ofthe 2D image. Such a method improves the design of a 3D modeled object.

Notably, the method allows the design modification of a 3D modeledobject in the context of a base mesh associated to a subdivision surfaceand to a corresponding predetermined mesh-to-NURBS-surface conversionalgorithm, with the advantages of such a design context well-known fromthe field of CAD. Indeed, the structure of the base mesh and theassociated user-intent regarding the design of the surface are keptthroughout the method, which simply amounts to adding a specification tothis base mesh framework to which CAD designers are used. Unlike priorart modification solutions performed on the base mesh (e.g. the additionof multi-resolution information and/or the addition of details aftersubdivisions), the method does not contemplate a subdivision of the basemesh before its later conversion (with the predeterminedmesh-to-NURBS-surface conversion algorithm at S30), such that said laterconversion may be performed easily (in terms of computational costs) andwill not result in a prohibitive number of control points. In otherwords, the number of faces of the mesh to be later inputted to theconversion algorithm is not unduly increased by the method. Unlike priorart modifications solutions performed on a NURBS, the method allows themodification of a model provided as a base mesh of a subdivisionsurface, such that the user may later perform other modifications, in arelatively simplified way, thanks to this context (with advantagesrelated to the arbitrary topology possible in this framework, and to thelittle memory space taken by a base mesh relative to a complex NURBS).Indeed, as widely known in the field of CAD, styling design is ratherperformed on subdivision surfaces than on NURBS for these reasons. Theuser may thus perform base mesh editions of the 3D modeled object evenafter the addition of the specification defined at S20. Also, the methodensures that continuity between NURBS patches is not unduly broken, asthe defining S20 is performed at the base mesh level.

The method thus offers to the user an easy, interactive and visual wayto add a local detail on a 3D modeled object, by defining aspecification at S20 in the form of a 2D image provided with a locationon the subdivision surface. This edition simulates a real engraving, andit respects the industrial requirements of the user, which includekeeping the context of a design based on the surface subdivisiontechnology (with a base mesh), where, in addition, continuity of thefinal surface matters.

The method is computer-implemented. Steps of the method are indeedexecuted on a computer system, by the computer system alone or bygraphical user-interaction (i.e. a user interacting with a GUI of thecomputer system). Thus, steps performed specifically by the computersystem may be performed fully automatically. In examples, the triggeringof at least some of the steps of the method may be performed throughuser-computer interaction. Notably, the defining steps S10 and S20 areperformed by the user, working on the computer, in the same work sessionor in different work sessions. The determining S30 may be triggered bythe user, in the same work session (the one of both S10 and S20 or theone of S20 only) or in a different work session. The level ofuser-computer interaction required may depend on the level of automatismforeseen and put in balance with the need to implement the user'swishes. In examples, this level may be user-defined and/or pre-defined.

A typical example of computer-implementation of the method is to performthe method with a system adapted for this purpose. The system maycomprise a processor coupled to a memory and a graphical user interface(GUI), the memory having recorded thereon a computer program comprisinginstructions for performing the method. The memory may also store adatabase. The memory is any hardware adapted for such storage, possiblycomprising several physical distinct parts (e.g. one for the program,and possibly one for the database).

The method generally manipulates modeled objects. A modeled object isany object defined by data stored in the database. By extension, theexpression “modeled object” designates the data itself “Designing a 3Dmodeled object” designates any action or series of actions which is atleast part of a process of elaborating a 3D modeled object. Thus, themethod may comprise creating the 3D modeled object from scratch.Alternatively, the method may comprise providing a 3D modeled objectpreviously created, and then modifying the 3D modeled object, notably bythe engraving achieved by the method.

According to the type of the system, the modeled objects may be definedby different kinds of data. The system may indeed be any combination ofa CAD system, a CAE system, a CAM system, a PDM system and/or a PLMsystem. In those different systems, modeled objects are defined bycorresponding data. One may accordingly speak of CAD object, PLM object,PDM object, CAE object, CAM object, CAD data, PLM data, PDM data, CAMdata, CAE data. However, these systems are not exclusive one of theother, as a modeled object may be defined by data corresponding to anycombination of these systems. A system may thus well be both a CAD andPLM system, as will be apparent from the definitions of such systemsprovided below.

By CAD system, it is meant any system adapted at least for designing amodeled object on the basis of a graphical representation of the modeledobject, such as CATIA. In this case, the data defining a modeled objectcomprise data allowing the representation of the modeled object. A CADsystem may for example provide a representation of CAD modeled objectsusing edges or lines, in certain cases with faces or surfaces. Lines,edges, or surfaces may be represented in various manners, e.g.non-uniform rational B-splines (NURBS). Specifically, a CAD filecontains specifications, from which geometry may be generated, which inturn allows a representation to be generated. Specifications of amodeled object may be stored in a single CAD file or multiple ones. Thetypical size of a file representing a modeled object in a CAD system isin the range of one Megabyte per part. And a modeled object maytypically be an assembly of thousands of parts.

In the context of CAD system, a modeled object may typically be a 3Dmodeled object, e.g. representing a product such as a part or anassembly of parts, or possibly an assembly of products. By “3D modeledobject”, it is meant any object which is modeled by data allowing its 3Drepresentation. A 3D representation allows the viewing of the part fromall angles. For example, a 3D modeled object, when 3D represented, maybe handled and turned around any of its axes, or around any axis in thescreen on which the representation is displayed. This notably excludes2D icons, which are not 3D modeled. The display of a 3D representationfacilitates design (i.e. increases the speed at which designersstatistically accomplish their task). This speeds up the manufacturingprocess in the industry, as the design of the products is part of themanufacturing or modeling process. The 3D modeled object further has theparticularity of being not only defined by the base mesh of S10, butalso by the data defined at S20 which form an additional specificationto those of traditional 3D modeled objects.

The 3D modeled object may represent the geometry of a product to bemanufactured in the real world subsequent to the completion of itsvirtual design with for instance a CAD software solution or CAD system,such as a (e.g. mechanical) part or assembly of parts, or more generallyany rigid body assembly (e.g. a mobile mechanism). A CAD softwaresolution allows the design of products in various and unlimitedindustrial fields, including: aerospace, architecture, construction,consumer goods, high-tech devices, industrial equipment, transportation,marine, and/or offshore or transportation. The 3D modeled objectdesigned by the method thus represents an industrial product which maybe a part of a terrestrial vehicle (including e.g. car and light truckequipment, racing cars, motorcycles, truck and motor equipment, trucksand buses, trains), a part of an air vehicle (including e.g. airframeequipment, aerospace equipment, propulsion equipment, defense products,airline equipment, space equipment), a part of a naval vehicle(including e.g. navy equipment, commercial ships, offshore equipment,yachts and workboats, marine equipment), a mechanical part (includinge.g. industrial manufacturing machinery, heavy mobile machinery orequipment, installed equipment, industrial equipment product, fabricatedmetal product, tire manufacturing product), an electro-mechanical orelectronic part (including e.g. consumer electronics, security and/orcontrol and/or instrumentation products, computing and communicationequipment, semiconductors, medical devices and equipment), a consumergood (including e.g. furniture, home and garden products, leisure goods,fashion products, hard goods retailers' products, soft goods retailers'products), a packaging (including e.g. food and beverage and tobacco,beauty and personal care, household product packaging). The method isparticularly useful where aesthetic aspects of the 3D modeled object areof high importance. This is true when the 3D modeled object represent avehicle or a plane mechanical part, such as visible part, or a consumergood (e.g. an electronic good, such as a camera).

By PLM system, it is meant any system adapted for the management of amodeled object representing a physical manufactured product. In a PLMsystem, a modeled object is thus defined by data suitable for themanufacturing of a physical object. These may typically be dimensionvalues and/or tolerance values. For a correct manufacturing of anobject, it is indeed better to have such values.

CAM stands for Computer-Aided Manufacturing. By CAM solution, it ismeant any solution, software of hardware, adapted for managing themanufacturing data of a product. The manufacturing data generallyincludes data related to the product to manufacture, the manufacturingprocess and the required resources. A CAM solution is used to plan andoptimize the whole manufacturing process of a product. For instance, itcan provide the CAM users with information on the feasibility, theduration of a manufacturing process or the number of resources, such asspecific robots, that may be used at a specific step of themanufacturing process; and thus allowing decision on management orrequired investment. CAM is a subsequent process after a CAD process andpotential CAE process. Such CAM solutions are provided by DassaultSystemes under the trademark DELMIA®.

CAE stands for Computer-Aided Engineering. By CAE solution, it is meantany solution, software or hardware, adapted for the analysis of thephysical behavior of a modeled object. A well-known and widely used CAEtechnique is the Finite Element Method (FEM) which typically involves adivision of a modeled objet into elements which physical behaviors canbe computed and simulated through equations. Such CAE solutions areprovided by Dassault Systemes under the trademark SIMULIA®. Anothergrowing CAE technique involves the modeling and analysis of complexsystems composed a plurality components from different fields of physicswithout CAD geometry data. CAE solutions allows the simulation and thusthe optimization, the improvement and the validation of products tomanufacture. Such CAE solutions are provided by Dassault Systemes underthe trademark DYMOLA®.

PDM stands for Product Data Management. By PDM solution, it is meant anysolution, software or hardware, adapted for managing all types of datarelated to a particular product. A PDM solution may be used by allactors involved in the lifecycle of a product: primarily engineers butalso project managers, finance people, sales people and buyers. A PDMsolution is generally based on a product-oriented database. It allowsthe users to share consistent data on their products and thereforeprevents actors from using divergent data. Such PDM solutions areprovided by Dassault Systemes under the trademark ENOVIA®.

FIG. 3 shows an example of the GUI of the system, wherein the system isa CAD system.

The GUI 2100 may be a typical CAD-like interface, having standard menubars 2110, 2120, as well as bottom and side toolbars 2140, 2150. Suchmenu- and toolbars contain a set of user-selectable icons, each iconbeing associated with one or more operations or functions, as known inthe art. Some of these icons are associated with software tools, adaptedfor editing and/or working on the 3D modeled object 2000 displayed inthe GUI 2100, displayed 3D modeled object 2000 being for example theresult of performing the method. The software tools may be grouped intoworkbenches. Each workbench comprises a subset of software tools. Inparticular, one of the workbenches is an edition workbench, suitable forediting geometrical features of the modeled product 2000. In operation,a designer may for example pre-select a part of the object 2000 and theninitiate an operation (e.g. change the dimension, color, etc.) or editgeometrical constraints by selecting an appropriate icon. For example,typical CAD operations are the modeling of the punching or the foldingof the 3D modeled object displayed on the screen. The GUI may forexample display data 2500 related to the displayed product 2000. In theexample of FIG. 2, the data 2500, displayed as a “feature tree”, andtheir 3D representation 2000 pertain to a brake assembly including brakecaliper and disc. The GUI may further show various types of graphictools 2130, 2070, 2080 for example for facilitating 3D orientation ofthe object, for triggering a simulation of an operation of an editedproduct or render various attributes of the displayed product 2000. Acursor 2060 may be controlled by a haptic device to allow the user tointeract with the graphic tools.

FIG. 4 shows an example of the system, wherein the system is a clientcomputer system, e.g. a workstation of a user.

The client computer of the example comprises a central processing unit(CPU) 1010 connected to an internal communication BUS 1000, a randomaccess memory (RAM) 1070 also connected to the BUS. The client computeris further provided with a graphical processing unit (GPU) 1110 which isassociated with a video random access memory 1100 connected to the BUS.Video RAM 1100 is also known in the art as frame buffer. A mass storagedevice controller 1020 manages accesses to a mass memory device, such ashard drive 1030. Mass memory devices suitable for tangibly embodyingcomputer program instructions and data include all forms of nonvolatilememory, including by way of example semiconductor memory devices, suchas EPROM, EEPROM, and flash memory devices; magnetic disks such asinternal hard disks and removable disks; magneto-optical disks; andCD-ROM disks 1040. Any of the foregoing may be supplemented by, orincorporated in, specially designed ASICs (application-specificintegrated circuits). A network adapter 1050 manages accesses to anetwork 1060. The client computer may also include a haptic device 1090such as cursor control device, a keyboard or the like. A cursor controldevice is used in the client computer to permit the user to selectivelyposition a cursor at any desired location on display 1080. In addition,the cursor control device allows the user to select various commands,and input control signals. The cursor control device includes a numberof signal generation devices for input control signals to system.Typically, a cursor control device may be a mouse, the button of themouse being used to generate the signals. Alternatively or additionally,the client computer system may comprise a sensitive/touch pad, and/or asensitive/touch screen.

The computer program may comprise instructions executable by a computer,the instructions comprising means for causing the above system toperform the method. The program may be recordable on any data storagemedium, including the memory of the system. The program may for examplebe implemented in digital electronic circuitry, or in computer hardware,firmware, software, or in combinations of them. The program may beimplemented as an apparatus, for example a product tangibly embodied ina machine-readable storage device for execution by a programmableprocessor. Method steps may be performed by a programmable processorexecuting a program of instructions to perform functions of the methodby operating on input data and generating output. The processor may thusbe programmable and coupled to receive data and instructions from, andto transmit data and instructions to, a data storage system, at leastone input device, and at least one output device. The applicationprogram may be implemented in a high-level procedural or object-orientedprogramming language, or in assembly or machine language if desired. Inany case, the language may be a compiled or interpreted language. Theprogram may be a full installation program or an update program.Application of the program on the system results in any case ininstructions for performing the method.

The defining S10 is now discussed.

As widely known from the field of CAD, a user may often define (i.e.design) a mesh, called “base mesh” associated to a subdivision surface.This is what is done at S10 in the method. A mesh designates any graphstructure, with vertices (3D positions) and edges (e.g. segments)linking vertices two-by-two and forming so-called faces of the mesh(i.e. smallest cycles of edges). The mesh may be one of different types,including quad meshes (consisting mainly or fully of four edge faces)and triangular meshes (consisting mainly or fully of three edge faces).The defining S10 may be performed in any way, including quickreproductions according to symmetry operations as in U.S. Pat. No.7,893,937 B2.

A base mesh in CAD is associated to a subdivision surface and to acorresponding predetermined mesh-to-surface conversion algorithm.Indeed, the base mesh is the underlying mesh in the data defining thegeometry of the 3D modeled object. In other words, the geometry of theobject is represented by the mesh. More specifically and as widelyknown, the base mesh is conceptually associated to a subdivisionalgorithm. Subdividing infinitely the base mesh according to saidsubdivision algorithm would result in a surface, called “subdivisionsurface”, that represents the envelope/boundary of the designed 3Dmodeled object and thereby models the 3D modeled object. It is thus saidthat the base mesh “controls” the subdivision surface. The subdivisionalgorithm may be any known subdivision algorithm, such as theCatmull-Clark subdivision algorithm in the case of a quad mesh, or theLoop subdivision algorithm in the case of a triangular mesh.

As widely known, for the purpose of practicability, the subdivisionsurface is actually not determined by the system. Instead, the systemdetermines an approximation thereof under the form of a set ofparametric patches, typically NURBS (e.g. Bézier) surface patches. Forexample, the system stores a program for performing a scheme thatconverts the mesh into a set of parametric patches that approximates thesubdivision surface. In this sense, the scheme corresponds to theunderlying subdivision algorithm. In the context of the method, this allmeans that the method is run on a system that supports subdivisionsurface technology, being notably configured/adapted for a user todefine a base mesh (such as at S10) and to automaticallyoutput/compute/determine a surface that approximates the real andtheoretical subdivision surface, according to a predeterminedsubdivision scheme (e.g. Catmull-Clark for a quad mesh or Loop for atriangular mesh), based on a predetermined conversion algorithm, forexample for display/editions/simulations or any other processing of thesurface known in CAD. By “approximation”, it is meant that the scalarvolume defined (integral of a positive scalar function) between thetheoretical subdivision surface and the output of the mesh-to-surfaceconversion algorithm is less than 10% or 5% of the volume of thesmallest parallelepiped bounding box enveloping the subdivision surface.

In the case of the method, the conversion algorithm predetermined to thesystem is a mesh-to-NURBS-surface conversion algorithm, meaning that thebase mesh is transformed in a set of NURBS elementary surfaces/patchesthat approximate the subdivision surface associated to the base mesh(the base mesh being a quad mesh and the subdivision scheme theCatmull-Clark subdivision scheme, in an example extensively illustratedlater). The predetermined mesh-to-NURBS-surface conversion algorithm maybe called “approximation operator” and it may be noted PCCM(•). Theconversion/transformation of a base mesh B₀ is written S₀=PCCM(B₀) inthe following. In an example, operator PCCM(•) is themesh-to-NURBS-surface conversion algorithm described in documents U.S.Pat. No. 7,595,799 B2 (corresponding to EP 1750229 B1) and U.S. Pat. No.7,400,323 B2 (where U.S. Pat. No. 7,595,799 B2 describes how to generateNURBS patches and U.S. Pat. No. 7,400,323 B2 how to improve quality ofthe geometrical connections between the patches), which are incorporatedherein by reference, in the configuration where it achieves a C²continuity everywhere (i.e. between all pairs of NURBS patches) exceptin the neighborhood of extraordinary vertices where it achieves at leastC⁰. But the predetermined mesh-to-NURBS-surface conversion algorithm maybe any other known operator producing untrimmed patches with C²continuity between each other except in the neighborhood ofextraordinary vertices where they are at least C⁰ (Catmull-Clarksubdivision convergence property, see Recursively generated B-splinesurfaces on arbitrary topological meshes, E. Catmull, J. Clark,Computer-Aided Design Volume 10, Issue 6, November 1978, Pages 350-355).For the sake of completeness, it is noted, as known, that such operatorsperform an initial number of subdivisions in order to have a homogenousmesh (usually two), and then create NURBS patches according to anequation solving, the complexity of the operation being increasinglydepending on the number of inputted mesh faces. Thereby, such operatorsare incompatible with multi-resolution meshes (meshes with localinformation to modify the predetermined subdivision scheme—so as to addlocal details), at least when the multi-resolution setting consists inadding information beyond the second or third level of subdivision.

As a reminder, a vertex is said to be regular (other vertices being onthe contrary said to be extraordinary) if they satisfy one of the twofollowing conditions:

-   -   The vertex does not lie on a free border of the mesh, and it is        surrounded by four non-sharp edges (and thus surrounded by four        faces).    -   The vertex lies on a free border of the mesh, and it is        surrounded by two free edges and one non-free and non-sharp edge        (thus surrounded by two faces).

This definition may also be found in the literature, notably in U.S.Pat. No. 7,595,799 B2 (corresponding to EP 1750229 B1) and U.S. Pat. No.7,400,323 B2. Regular vertices ensure the best surface quality inCatmull-Clark subdivision theory.

As known per se, a NURBS surface is defined by a (2D) grid of so-called“control points”, with the usual topological definition of “grid”, e.g.with several (at least two) rows and several (at least two) columns.Throughout the method, the computer system determines S20, a grid ofsurface points that belong to the NURBS surface and that corresponds tothe grid of control points according to the predetermined invertiblefunction. The surface points are points geometrically lying on/belongingto the surface. In the case of a NURBS, a straightforward grid ofsurface is the grid of Gréville points of the example. The predeterminedinvertible function that allows the one-to-one correspondence betweensurface Gréville points and control points (generally outside thesurface itself) of a NURBS is predetermined and provided from the widelyknown mathematics describing NURBS geometry. These mathematics arehowever detailed hereunder for the sake of completeness. It is notedthat the determining S20 may be performed in any way, e.g. as abackground process that re-computes real-time the Gréville points eachtime the surface and its control points are modified.

A NURBS surface is a surface, widely used in CAD, defined in a 2D spacewith values in a 3D space: S(u, v)→(x, y, z). A NURBS surface isrepresented with two knot vectors and a 2-dimensional array of controlpoints, and possibly other data depending on the CAD software used.Typically, the knot vectors are 1D arrays of floating point values. Thearray of control points may be indexed. Each control point is a 3D pointwith coordinates x, y, z. The place in the array is defined with the twoindices i and j; with 0≦i<M and 0≦j<N. A set of parameters known as theGréville parameters can be computed from the knot vectors so that eachvalue of the indices i or j can be associated to a Gréville parameter(u_(i), v_(j)). Traditionally, the Gréville parameters are a couple ofmeans of a fixed number of consecutive values of the knot vector. Thisfixed number, called “degree”, may be different for each dimension (u orv) of the knot vector. In general, a couple of degrees belong to thedata representing the NURBS surface. For more details, the article“Curves and surfaces for Computer Aided Geometric Design”, ed. MorganKaufmann, (2001) by Gerald Farin provides the basics of NURBS surfaces.

Classically, a NURBS surface is defined by the following inputs.

-   -   1. A grid of (n+1) (m+1) control points P_(ij)ε        ³ with iε{0, . . . , n} and jε{0, . . . , m}.    -   2. Two positive integer numbers p,q named the “degrees” for some        reasons explained later.    -   3. A first ordered list of real numbers u₀≦ . . .        ≦u_(i)≦u_(i+1)≦ . . . ≦u_(p+n+1) named the “u knot vector” such        that the first p+1 values u_(i) are equal and such that the last        p+1 values u_(i) are equal.    -   4. A second ordered list of real numbers v₀≦ . . .        ≦v_(j)≦v_(j+1) . . . ≦v_(q+m+1) named the “v knot vector” such        that the first q+1 values v_(j) are equal and such that the last        q+1 values v_(j) are equal.    -   5. A grid of (n+1) (m+1) weight numbers ω_(ij)>0 with iε{0, . .        . , n} and jε{0, . . . , m}.

These inputs define two B-spline basis, respectively noted N_(i)^(p):[u₀,u_(p+n+1)]→

⁺ with iε{0, . . . , n} and N_(j) ^(q):[v₀,v_(q+m+1)]→

⁺ with jε{0, . . . , m}. Basis B-spline functions N_(i) ^(p) and N_(j)^(q) are piecewise polynomial with respective degrees p and q. Theirdetailed definition is not useful to the invention. The NURBS surfaceS:[u₀,u_(p+n+1)]×[v₀,v_(q+m+1)]→

³ is defined by the following rational parameterization.

${S\left( {u,v} \right)} = {\frac{1}{\sum\limits_{i = 0}^{n}\; {\sum\limits_{j = 0}^{m}\; {\omega_{ij}{N_{i}^{p}(u)}{N_{j}^{q}(v)}}}}{\sum\limits_{i = 0}^{n}\; {\sum\limits_{j = 0}^{m}\; {\omega_{ij}P_{ij}{N_{i}^{p}(u)}{N_{j}^{q}(v)}}}}}$

Similarly, a NURBS curves C:[t₀,t_(p+n+1)]→

³ is defined by a degree p a list of n+1 control points P_(i) andweights ω_(i) involved in the following formula.

${C(t)} = {\frac{1}{\sum\limits_{i = 0}^{n}\; {\omega_{i}{N_{i}^{p}(t)}}}{\sum\limits_{i = 0}^{n}\; {\omega_{i}P_{i}{N_{i}^{p}(t)}}}}$

In fact, from the mathematical point of view, a NURBS surface is thetensor product of two NURBS curves. For clarity, the method is mainlyillustrated (on the following figures) with NURBS curves rather thanNURBS surfaces.

Gréville points are now briefly discussed.

Gréville abscissas are particular values of (u,v) parameters noted(u_(k)*,v_(l)*) with kε{0, . . . , n} and lε{0, . . . , m}. They areuniquely defined by the following formulas.

$u_{k}^{*} = {{\frac{1}{p}{\sum\limits_{\sigma = k}^{k + p - 1}\; {u_{\sigma}\mspace{14mu} v_{1}^{*}}}} = {\frac{1}{q}{\sum\limits_{\sigma = 1}^{1 + q - 1}\; v_{\sigma}}}}$

FIG. 5 illustrates a NURBS curve together with its n+1=6 control pointsP_(i) and Gréville points G_(i).

It is noted that the predetermined conversion algorithm may compriserefinements (e.g. ulterior processing) that locally modify or erase theNURBS structure (possibly according to a local specificity of the basemesh or to a specification added to the base mesh). These include forexample smoothening of portions as in U.S. Pat. No. 7,952,575 B2, ordefinition of character lines as in US 2013293541 A1. Nevertheless, thesurface consists in the end (for at least 90%—if not 100%—of its area)in NURBS. It is noted that the method forms itself such a refinement. Itis also noted that concurrence between such refinements and the localchange specification involved in the present method may be handled inany way, being merely an implementation detail. It is also noted thatthe implementation of the method may be performed in any way, such thatthe order between the refinements, and the full sequence of processingis not relevant. This is encompassed by the fact that it is said thatthe NURBS surface determined at S30 “corresponds” to performing a givenprocessing. The NURBS determined at S30 may thus be submitted to suchfurther refinement processing and still correspond to the mainprocessing addressed in S30. In an example however, the base mesh isprovided with no additional specification (other than the one of S20),i.e. the base mesh is “raw”. In another example, the concurrentrefinements affect the whole surface for less than 10% of its area, oreven less than 5% of its area, such that there is an exact match, forthe complementary area, between the result determined at S30 and theprocessing mentioned.

The defining S10 is performed by the user. This results in a base meshwhich achieves a good trade-off between description of the surface andnumber of points. This is unlike mere subdivisions. The base meshdefined at S10 may be different from a mesh stemming from the meresubdivision of another mesh (e.g. according to the associatedsubdivision scheme). If such subdivision is invoked by the user duringthe design process, at least one vertex resulting from the subdivisionmay be manually moved (i.e. position directly—e.g. graphically, bydrag-and-drop—specified by the user). All in all, starting from scratch,the defining S10 may comprise a number of subdivisions inferior to 5, orto 2. Such subdivisions allow the definition of details and thus complexdesign, but a base mesh is a mesh designed by a user, such that it isnot too complex (relatively speaking), and such that its conversion withthe above-mentioned category of conversion algorithms is relativelyquick and leads to a reasonable number of NURBS control points. Forexample, considering the total length L of the 3D modeled object (e.g.largest length of the smallest parallelepiped bounding box envelopingthe 3D modeled object), the average length of an edge of a base meshdefined at S10 by the user—to represent the 3D modeled object—may behigher than 1% of L, 5% of L or even 10% of L. These features are keptin the end of the method, which saves user-intent.

The defining S20 is now discussed.

Again, this defining S20 is performed by the user. The data inputted bythe user at this point may be kept in the model as an additionalspecification of the 3D modeled object, which is thus at least definedby the base mesh (associated to a subdivision algorithm/surface and to apredetermined operator noted as above PCCM( ), which are predeterminedto the system), and also by a 2D image and a location for engraving the2D image on the subdivision surface. The defining S20 thus correspondsto the setting of a feature, in a feature-based CAD system, and/or thesetting of a historical design operation, i.e. a node in a history treeof a history-based CAD system, such systems being known per se. The 2Dimage is any type of image. It may be a photograph (e.g. RGB orgrayscale), or a 2D picture drawn by a user—the same user or anotheruser (e.g. again, RGB or grayscale). Typically, the 2D represents alocal detail to be engraved on the final designed and/or manufacturingobject, such as a logo. The aim of the method is to perform an“engraving” design operation, which is defined by the result of thedetermining S30 as such and visually represents (when then displayingthe surface outputted by the determining S30) a real-world engraving ofthe design contained in the 2D view onto the subdivision surface.However, this is internal to the system, which is predetermined toimplement S30 (e.g. upon user-request/user-triggering). The user onlyspecifies the location to which the 2D image should be applied. The usermay specify such a location on the subdivision surface in any way, forexample via a mere isomorphic link between the 2D image and thesubdivision surface.

For example, the base mesh being associated to the subdivision surface,and the system being configured to compute an approximation thereof(thanks to operator PCCM( )), the method may comprise displaying said(approximation of) the subdivision surface. The user may thengraphically define the location. For example, the user may position a 3Drepresentation of the 2D image anywhere on (next to) the subdivisionsurface (the base mesh being 3D, the 2D image may be moved in a 3Dworkspace containing the 3D base mesh), and the size of the 2D image maypotentially be increased or reduced at will, and the system may performa predetermined projection (of the 2D image onto the surface, or of thesurface onto the 2D image). The user may also specify the orientation ofthe 2D image. The user may also specify the projection. Anyuser-friendly and graphical way of performing such positioning (whichmay amount to projecting/sticking/applying the 2D image onto arepresentation of the subdivision surface) may be implemented by themethod.

The location provided at S20 by the user is in any case interpreted bythe system as a location on the subdivision surface, and thus impactingunderlying faces of the base mesh or NURBS patches. The system may thusassociate faces of the base mesh and/or NURBS patches to the location.Indeed, the predetermined conversion algorithm associates faces of thebase mesh to resulting NURBS patches. Thus, when the user locates the 3Dimage on the subdivision surface, the user actually locates it on asubset of the NURBS patches, and therefore, a corresponding subset ofbase mesh faces. This association/impact retrieval may be implemented inany way. The method may be for adding small details to the subdivisionsurface, while saving its overall shape. First, the topology of thesurface is saved. Also, more than 75% of the initial subdivision surface(i.e. the result of simply converting the base mesh), or more than 90%may be left unmodified, the size of the 2D image (e.g. after thepotential expansion) being adapted for that.

The determining S30 is now discussed.

The method determines a NURBS surface which is different from thesurface outputted by the predetermined mesh-to-NURBS-surface conversionalgorithm (i.e. the PCCM( ) operator), which may be called “raw result”or “raw NURBS/surface”. The NURBS determined at S30 is actually one thatcorresponds to applying a deformation map on the result of performingthe mesh-to-NURBS-surface conversion algorithm to the base mesh (whichis not necessarily done). Thus, the deformation map (at leastconceptually) modifies the “raw result” to engrave the surface with the2D image. For that, the deformation map includes, by definition,displacement vectors provided for positions of said raw result, and notany positions, but positions that correspond to the location forengraving the 2D image. And the displacement vectors are computedprecisely based on corresponding pixel values of the 2D image, such thatit is indeed the 2D image which is engraved on the subdivision surface.It is noted that the displacement vectors may be null (i.e. nodisplacement). Actually, the deformation map may be defined for a widerdomain than the location for engraving the 2D image, e.g. the wholesurface, as long as the displacement vectors are null where appropriate(notably at locations not concerned by the engraving). This is a merematter of implementation. In short, the deformation map is a functionthat specifies a 3D motion of positions of the subdivision surface (andthus positions of the underlying NURBS patches, such as control pointsthereof, and/or positions on the faces of the base mesh)), the methodimplementing a predetermined association (or computation thereof, i.e.the value is predetermined, or the way of computing it is predeterminedand does not depend on any further user input) between said positionsand pixels of the 2D image (e.g. pixels being retained depending on agiven criterion), and the method further implementing a predeterminedcomputation of the 3D motion value based on the associated pixels. It isnoted that the direction of the 3D motion (or computation thereof, i.e.the value is predetermined, or the way of computing it is predeterminedand does not depend on the 2D image or any further user input, but forexample only on the base mesh) may be predetermined.

The displacement vectors are computed in any way, as long as it is basedon the 2D image, with the function of forming an engraving feature. Thepixel values of the image are mapped to positions of the subdivisionsurface (thanks to the defining step S20), and thus to positions of theconsidered NURBS surface. In an example, the pixel values contemplatedby the method are pixel grayscale values. In other words, a grayscalevalue associated to pixels of the 2D image is determined, and used tocompute the displacement vector. The 2D image may be provided ingrayscale, in which case the grayscale value may be directly the valueof a pixel. The 2D image may also be provided in RGB, in which case themethod may use any predetermined RGB-grayscale conversion algorithm. Inboth cases, the method may implement any further processing on valuesretained.

In an example, the amplitude (or magnitude) of a displacement vectorincreasingly depends on the corresponding grayscale value. In otherwords, the higher the (grayscale) value of a retained pixel value, themore the method modifies (in amplitude) the raw surface. Any increasingfunction may be implemented. In an example, the function is simply anindicator function (displacement if the value of the pixel is above apredetermined threshold, no displacement otherwise, with possibly asmoothening between “positive displacement” values and “no displacement”values in order to avoid brutal transitions). In another example, thefunction is a more refined function, such as a linear function.

Also, the user may further select an engraving direction from apredetermined binary list. Typically, the user projects/sticks manuallythe 2D image onto the surface, as explained earlier, and the user maythen select if the displacement should be in the direction of projection(i.e. from the user to the surface) or in the backward direction (i.e.from the surface to the user). Alternatively, the engraving directionmay be predetermined as mentioned above. These are all configurablefeatures that increase the user-friendliness of the system.

A specific example of the method will now be discussed with reference toFIGS. 6-19.

As shown on the flowchart of FIG. 6, in the example the method actuallyperforms S30 by computing the displacement vectors (and thus thedeformation map) at S32 and in parallel by performing at S34 themesh-to-NURBS-surface conversion algorithm to the base mesh, noted B₀,and then the method of the example actually applies at S36 thedeformation map (noted i,j

Δ(i,j)) on the result (noted S₀=PCCM(B₀)) of S34.

In specific, the computing S32 includes first determining S320, from B₀,a first 3D mesh (B₄=R^(n)(B₁)) that corresponds to the location forengraving the 2D image. The first 3D mesh, noted B₄, corresponds to apart of B₀ where the 2D image is to be engraved, according to the userspecifications of S20. It may correspond to all faces of B₀ impacted bythe defining S20 (a neighborhood related to the location defined at S20,which may be determined in any predetermined way) and may be asubdivision thereof. It is noted that a strip of non-impacted faces mayoptionally be also retained.

The computing S32 then includes determining S322, from B₄, a second 3Dmesh (B₅(i)=B₄(i)+d_(i)N(i)) by displacing vertices of B₄, in adirection normal to the B₄, based on the corresponding pixel values ofthe 2D image. In other words, the method contemplates the part of thebase mesh impacted by the engraving and displaces positions thereofaccording to the pixel values of the engraving image. The direction ofdisplacement is computed in a predetermined way, as a normal directionof a mesh at vertices (any normal direction computation may beimplemented). This may be done according to the above explanations, e.g.the pixel grayscale value determininglinearly/proportionally/increasingly the amplitude of displacement (thedirection being normal, the orientation—toward the user orbackward—being user-selectable or predetermined).

There, the computing S32 may compute S324, for positions of the resultof S34, a displacement vector (Δ(i,j)=C_(i,j)(S₅)−C_(i,j)(S₄)) equal tothe 3D position difference between the corresponding control point(C_(i,j)(S₅)) of the result (S₅=PCCM(B₅)) of performing the mesh-to-NURBS-surface conversion algorithm to B₅ and the corresponding control point(C_(i,j)(S₄) of the result (S₄=PCCM(B₄)) of performing themesh-to-NURBS-surface conversion algorithm to B₄. Here, S₄ and S₅ may becomputed to perform that quickly. But any other computationimplementation may be contemplated.

The specific example based on FIG. 6 performs particularly efficiently,thanks to the specific computations it performs. Moreover, this exampleensures that the required continuity outputted by the PCCM( ) ispreserved (a C² continuity everywhere except in the neighborhood ofextraordinary vertices where it achieves at least C⁰). Indeed, thespecific algebra implemented by the example ensure this preservation, inrelation with the features of the PCCM operator.

As shown on FIG. 7 which reminds the overall context, the input data arethe base mesh of a subdivision surface and a 2-dimensional image. Thegoal is to engrave the details of the 2D image on the surface defined bythe base mesh. This engraving results in a local deformation. The 2-Dimage is positioned with respect to the surface in order to define wherethe details of the image are to be located on the surface. The outputdata is a NURBS surface featuring the engraved image. Outside theengraved details, the overall shape of the initial surface is unchanged.

The process is as follows:

-   -   1. By using the initial base mesh B₀, convert the input        subdivision surface into a NURBS surface S₀.    -   2. Perform base mesh computations by using the initial base mesh        B₀ and the 2D image. This provides NURBS surfaces S₄ and S₅.        Surface S₄ is the local unchanged shape and surface S₅ is the        local engraved shape.    -   3. From NURBS surfaces S₄ and S₅, compute control points        deviations.    -   4. Engrave the details of the 2D image by applying these control        points deviations on NURBS surface S₀. This yields the resulting        NURBS surface S₁.

Clearly, the method saves the best of both worlds (subdivision surfacedesign and NURBS design), meaning that details of an image can beengraved on a subdivision surface without changing the shape outside theengraved area. This accurately captures the design.

The whole process of the example is fully automatic and can be stored asan “engraving” feature in the data structure of a history based CADsystem. The inputs are the subdivision surface and the image, and theoutput is the engraved NURBS surface. After the engraving operation isperformed, the subdivision surface can be edited or the image definitioncan be changed as well. The modified version of the resulting surface isobtained by replaying the engraving feature. This allows easy and fastdesign changes.

Despite the resulting surface is a NURBS surface, from the user point ofview the functionality is that of a subdivision surface. This is becausethe edition is performed on the input subdivision surface. The NURBSaspect of the resulting surface is a way to handle the result that doesnot restrict edition capabilities.

The example is now discussed with more implementation details.

Input data and operators are detailed as follows. The subdivisionsurface defined by its base mesh B₀ is approximated by a NURBS surfaceS₀. FIG. 8 illustrates the base mesh B₀ of a 3D modeled object thatrepresents a vase. FIG. 9 illustrates the NURBS surface S₀ (what isdisplayed to final users, e.g. customers).

The plane including the image is noted P and its specifications aredetermined continuously e.g. updated) by the method. It is defined by apoint X₀ and two perpendicular normal vectors U,V. Any point in theplane can be written X₀+uU+vV, where (u,v)ε

². The position of plane P defines where the image is to be engraved onsurface S₀. The deformation map d:

²→

is defined according to the U,V vectors of plane P. It captures a greylevel associated to each image point and d(u,v)=0 means that there is noimage detail at point (u,v). The image is included in a rectangleI=[u_(min),u_(max)]×[v_(min),v_(max)] meaning that d(u,v)=0 outside thisrectangle. FIG. 10 illustrates rectangle I on plane P and itspositioning with respect to surface S₀. Function d may be an increasingfunction, this being an implementation detail to the skilled person.

The method of the example may implement a flattening operation from 3Dspace to plane P defined by a mapping Q:

³→P. Whether mapping Q is linear or not is not relevant. Orthogonalprojection from 3D space to plane P is defined by the linear mapping H:

³→P, that is:

H(X)=X ₀ +

X−X ₀ ,U

U+

X−X ₀ ,V

V

The mesh refinement of a mesh M_(i) into a finer mesh M_(i+1) is notedR, meaning that M_(i+1)=R(M_(i)). Typically, operator R is theCatmull-Clark subdivision step. Getting the control point P_(i,j) of aNURBS surface S is noted P_(i,j)=C_(i,j)(S).

The main phase of base mesh computations is now described. The diagramof FIG. 11 illustrates the whole process, and forms a detail of the“Base mesh computations” block of the diagram of FIG. 7.

The first step is to identify the faces of B₀ on which the image will beengraved. Let B₁ be the base mesh extracted from B₀, as illustrated byFIG. 12.

This mesh is flattened in plane P through the B₂=Q(B₁) operation,yielding the flat mesh B₂ featuring the same topology, as illustrated byFIG. 13.

Now, the determining S310 of the first 3D mesh (B₄) includes not onlyextracting a sub-mesh B₁ of the base mesh that corresponds to thelocation for engraving the 2D image, as previously shown, but it mayalso include then subdividing the sub-mesh B₁ a predetermined number oftimes n, as mentioned earlier. This allows adaptation to the details ofthe 2D image and a more refined engraving.

The predetermined number n may be defined by the user (interactively).But it may also increasingly depend on a predetermined criterionassociated to a level of details of the 2D image. In this latter case,the determination of n may be performed fully automatically, byevaluating the predetermined criterion.

An example of such automatism is provided below.

Mesh B₂ is to be subdivided until the mesh size is compatible with thedetails of the image. This yields mesh B₃=R^(n)(B₂), where integer n isthe number of subdivisions, as illustrated in FIG. 14. The samesubdivision process B₃=R^(n)(B₂) is then to be applied to the non-planarbase mesh B₁, that is B₄=R^(n)(B₁), as illustrated in FIG. 15. Meshes B₄and B₃ have the same topology. Mesh B₃ is planar, mesh B₄ is non-planar.

The subdivision level n is estimated by ensuring a sufficient number ofsubdivision points to represent the input image, in particular in itsnon-null deformation areas:

-   -   from B₂ it is easy to estimate the mean length between two        neighbor subdivision points.    -   from the input image, an erosion-like algorithm can compute the        mean thickness over all the non-null deformation areas in the        image space (see The Scientist and Engineer's Guide to Digital        Signal Processing, Steven W. Smith).

n may then be first chosen so (e.g. as the minimum integer that ensures)that the ratio

$\frac{{Mean}\mspace{14mu} {thickness}\mspace{14mu} {of}\mspace{14mu} {deformation}\mspace{14mu} {areas}}{3*\left( {{Mean}\mspace{14mu} {distance}\mspace{14mu} {between}\mspace{14mu} {subdivision}\mspace{14mu} {points}} \right)}$

is greater than 1. This criterion is a good indicator for the user.

For each integer i, let X_(i)=(x_(i),y_(i),z_(i)) be the 3D coordinatesof the i-th point B₃(i) of planar mesh B₃. In the local 2D axis systemof plane P, the 2D coordinates of this point are (u_(i),v_(i)) withu_(i)=

X_(i)−X₀,U

U and v_(i)=

X_(i)−X₀,V

V. Let d_(i)=d(u_(i),v_(i)) be the deformation map value at point(u_(i),v_(i)) of plane P.

The mesh B₄ is now changed into another mesh B₅ featuring the sametopology and modified as follows. For each integer i, let N(i) be thenormal vector computed on mesh R^(n-1) (B₁) by using neighboring pointsof B₄(i). Vector N(i) is computed by averaging the normal around eachsubdivision cell at level n−1. Then, the i-th point of mesh B₅ isdefined by B₅(i)=B₄(i)+d_(i)N(i). This is illustrated by FIG. 16.

So far, all computations are performed on meshes. Now, it is time totranslate the information in terms of control points because, in theend, a NURBS surface is the output. Let S₄ be the NURBS surface thatapproximates the subdivision surface defined by the mesh B₄, that isS₄=PCCM(B₄), as shown on FIG. 17. Similarly, let S₅ be the NURBS surfacethat approximates the subdivision surface defined by the mesh B₅, thatis S₅=PCCM(B₅), as shown on FIG. 18.

It must be understood that S₅ is not the local shape of the resultingsurface. The boundaries of S₅ do not fit the initial surface S₀.

In order to report the deformation on S₀, a new function Δ is computedfrom S₄ and S₅.

By construction:

-   -   The NURBS patches from S₄(resp S₅) are C² between each other        except in the neighborhood of extraordinary vertices where they        are at least C⁰ (from PCCM(•) definition)    -   Each base mesh face in the deformation area can be linked to two        NURBS patches: one in S₄ and one in S₅. However these two        patches do not have the same knot vectors in general and before        defining Δ the vectors may be homogenized for every couple of        patches by performing knot insertion or degree elevation (see        “The NURBS Book”, L. Piegl and T. Wayne, Springer, 2nd edition,        2013).

The map i,j

Δ(i,j) of control points deviations is defined by the difference of S₅and S₄ control points: Δ(i,j)=C_(i,j)(S₅)−C_(i,j)(S₄). This deformationmap is to be applied on S₀ only on the subset of patches of thedeformation area. Here again the knot vectors of these patches are notcompatible in general with the ones of Δ, so they have to be homogenizedtoo.

Once done, the deformation of the initial surface S₀ is obtained byapplying deviations Δ (i,j) to the control points of involved NURBSsurfaces, which yields the output surface S₁. The geometric continuityof S₁ is preserved by applying Δ. If φ represents the shape computedfrom the control points of Δ and the homogenized knot vectors of S₀, thepatches of φ are C² between each other except in the neighborhood ofextraordinary vertices where they are at least C⁰. Thus S₁ can beexpressed as the linear combination: S₁=S₀+S₅−S₄, S₀, S₄ and S₅ have thesame parameterization (their patches are sharing the same knot vectors),they are C² almost everywhere, so is S₁. As noted, the method mayimplemented appropriate above-mentioned homogenization (as widely knownand explained in see “The NURBS Book”, L. Piegl and T. Wayne, Springer,2nd edition, 2013) wherever necessary.

FIG. 19 shows the final result.

FIGS. 20-26 show applications of the method in real cases.

FIG. 20 shows different textures and FIG. 21 shows the result ofapplying such texture on a top face of parallelepiped 3D modeled object(modeled by an underlying base mesh not displayed to the user in themode of FIG. 21, which can be in this case, very simple), thanks to themethod. As can be seen, the method may be used to engrave a texture on aplane, in relief. FIG. 22 shows hairdryers with drawings engraved ontheir front face. FIG. 24 shows the result of engraving the patternshown on FIG. 23 on a plane and on a razor stick. FIG. 25 shows thefully designed razor, with said pattern engraved on it. FIG. 26 showsthe result of engraving logo 260 and pattern/relief 262 on a 3D modeledobject 264, with the method (object 264 being, again, modeled by anunderlying base mesh not displayed to the user in the mode of FIG. 21).

1. A computer-implemented method for designing a 3D modeled object, themethod comprising: defining, by a user, a base mesh (B₀) associated to asubdivision surface and to a corresponding predeterminedmesh-to-NURBS-surface conversion algorithm (PCCM(•)), the subdivisionsurface representing the 3D modeled object; defining, by the user, a 2Dimage and a location for engraving the 2D image on the subdivisionsurface; and determining a NURBS surface (S₁) that corresponds toapplying a deformation map (i,j

Δ(i,j)) on the result (S₀=PCCM(B₀)) of performing themesh-to-NURBS-surface conversion algorithm to the base mesh, thedeformation map including displacement vectors provided for positions ofthe result of performing the mesh-to-NURBS-surface conversion algorithmto the base mesh, the positions corresponding to the location forengraving the 2D image, the displacement vectors being computed based oncorresponding pixel values of the 2D image.
 2. The method of claim 1,wherein the method comprises computing the displacement vectors,including: determining, from the base mesh (B₀), a first 3D mesh(B₄=R^(n)(B₁)) that corresponds to the location for engraving the 2Dimage; determining, from the first 3D mesh, a second 3D mesh(B₅(i)=B₄(i)+d_(i)N(i)) by displacing vertices of the first 3D mesh, ina direction normal to the first 3D mesh, based on the correspondingpixel values of the 2D image; and computing, for positions of the resultof performing the mesh-to-NURBS-surface conversion algorithm to the basemesh, a displacement vector (Δ(i,j)=C_(i,j)(S₅)−C_(i,j)(S₄)) equal tothe 3D position difference between the corresponding control point(C_(i,j)(S₅)) of the result (S₅=PCCM(B₅)) of performing themesh-to-NURBS-surface conversion algorithm to the second 3D mesh and thecorresponding control point (C_(i,j)(S₄) of the result (S₄=PCCM(B₄)) ofperforming the mesh-to-NURBS-surface conversion algorithm to the first3D mesh.
 3. The method of claim 2, wherein determining the first 3D mesh(B₄=R^(n)(B₁)) includes extracting a sub-mesh (B₁) of the base mesh thatcorresponds to the location for engraving the 2D image, and thensubdividing the sub-mesh (B₁) a predetermined number of times (n). 4.The method of claim 3, wherein the predetermined number of times (n) isdefined by the user and/or increasingly depends on a predeterminedcriterion associated to a level of details of the 2D image.
 5. Themethod of claim 1, wherein the corresponding pixel values of the 2Dimage are pixel grayscale values.
 6. The method of claim 5, wherein theamplitude of a displacement vector increasingly depends on thecorresponding grayscale value.
 7. The method of claim 1, wherein theuser further selects an engraving direction from a predetermined binarylist.
 8. A non-transitory computer readable medium storing a computerfile including specifications of: a base mesh associated to asubdivision surface and to a corresponding predeterminedmesh-to-NURBS-surface conversion algorithm, the subdivision surfacerepresenting the 3D modeled object; and a 2D image and a location forengraving the 2D image on the subdivision surface.
 9. A non-transitorydata storage medium having recorded thereon a computer programcomprising instructions for performing a method for designing a 3Dmodeled object, the method comprising: defining, by a user, a base meshassociated to a subdivision surface and to a corresponding predeterminedmesh-to-NURBS-surface conversion algorithm, the subdivision surfacerepresenting the 3D modeled object; defining, by the user, a 2D imageand a location for engraving the 2D image on the subdivision surface;and determining a NURBS surface that corresponds to applying adeformation map on the result of performing the mesh-to-NURBS-surfaceconversion algorithm to the base mesh, the deformation map includingdisplacement vectors provided for positions of the result of performingthe mesh-to-NURBS-surface conversion algorithm to the base mesh, thepositions corresponding to the location for engraving the 2D image, thedisplacement vectors being computed based on corresponding pixel valuesof the 2D image.
 10. A CAD system comprising a processor coupled to aGUI and to a non-transitory memory, the non-transitory memory havingrecorded thereon a computer program comprising instructions forperforming a method for designing a 3D modeled object, the methodcomprising: defining, by a user, a base mesh associated to a subdivisionsurface and to a corresponding predetermined mesh-to-NURBS-surfaceconversion algorithm, the subdivision surface representing the 3Dmodeled object; defining, by the user, a 2D image and a location forengraving the 2D image on the subdivision surface; and determining aNURBS surface that corresponds to applying a deformation map on theresult of performing the mesh-to-NURBS-surface conversion algorithm tothe base mesh, the deformation map including displacement vectorsprovided for positions of the result of performing themesh-to-NURBS-surface conversion algorithm to the base mesh, thepositions corresponding to the location for engraving the 2D image, thedisplacement vectors being computed based on corresponding pixel valuesof the 2D image.